Tensor-product vertex patch smoothers for biharmonic problems

TL;DR

This paper introduces efficient tensor-product vertex patch smoothers for biharmonic problems, demonstrating high convergence rates and effective implementation strategies, including low-rank tensor approximations and mixed-precision computing.

Contribution

It presents novel vertex patch smoothers with tensor-product structures for fourth order PDEs, including implementation techniques and performance analysis.

Findings

High convergence rates for vertex patch smoothers

Multiplicative smoother requires fewer iterations

Additive smoother performs better in 3D due to parallelism

Abstract

We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations. However, in three-dimensional cases, the additive smoother outperforms its multiplicative counterpart due to the latter's lower potential for parallelism. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 percent.